Binary black hole coalescence in the extreme-mass-ratio limit: Testing and improving the effective-one-body multipolar waveform

We discuss the properties of the effective-one-body (EOB) multipolar gravitational waveform emitted by nonspinning black-hole binaries of masses $\mu$ and $M$ in the extreme-mass-ratio limit $\mu /M=\nu \ll 1$. We focus on the transition from quasicircular inspiral to plunge, merger, and ringdown. We compare the EOB waveform to a Regge-Wheeler-Zerilli waveform computed using the hyperboloidal layer method and extracted at null infinity. Because the EOB waveform keeps track analytically of most phase differences in the early inspiral, we do not allow for any arbitrary time or phase shift between the waveforms. The dynamics of the particle, common to both wave-generation formalisms, is driven by a leading-order $\mathcal{O}(\nu )$ analytically resummed radiation reaction. The EOB and the Regge-Wheeler-Zerilli waveforms have an initial dephasing of about $5\times {10}^{-4}\mathrm{rad}$ and maintain then a remarkably accurate phase coherence during the long inspiral ($\sim 33$ orbits), accumulating only about $-2\times {10}^{-3}\mathrm{rad}$ until the last stable orbit, i.e. $\Delta \varphi /\varphi \sim -5.95\times {10}^{-6}$. We obtain such accuracy without calibrating the analytically resummed EOB waveform to numerical data, which indicates the aptitude of the EOB waveform for studies concerning the Laser Interferometer Space Antenna. We then improve the behavior of the EOB waveform around merger by introducing and tuning next-to-quasicircular corrections in both the gravitational wave amplitude and phase. For each multipole we tune only four next-to-quasicircular parameters by requiring compatibility between EOB and Regge-Wheeler-Zerilli waveforms at the light ring. The resulting phase difference around the merger time is as small as $\pm 0.015\mathrm{rad}$, with a fractional amplitude agreement of 2.5%. This suggest that next-to-quasicircular corrections to the phase can be a useful ingredient in comparisons between EOB and numerical-relativity waveforms.

Figure 1

Multipolar “convergence” of the $\mathcal{R}{h}_{+}/(M\nu )$ polarization of the Regge-Wheeler-Zerilli multipolar waveform. Top panel: The complete wave train ($\sim 37$ cycles). Bottom panel: Impact of subdominant multipoles around the merger time. The vertical dashed line indicates the light-ring crossing.

Figure 2

Testing the waveform resummation for $\ell =2$ at the beginning of the inspiral. Top panel: Insplunge and RWZ waveforms extracted at ${\mathcal{I}}^{+}$. Bottom panel: The phase differences $\Delta {\varphi }_{\ell m}^{\mathrm{EOBRWZ}}={\varphi }_{\ell m}^{\mathrm{EOB}}-{\varphi }_{\ell m}^{\mathrm{RWZ}}$, for RWZ waveforms measured at ${\mathcal{I}}^{+}$, are contrasted with the corresponding ones for RWZ waveforms measured at a finite extraction radius $r_{*}^{\mathrm{obs}}=1000M$.

Figure 3

Addition of NQC corrections and matching to QNMs; $\ell =2$ multipoles. The (light) dashed lines refer to the bare insplunge waveform, without the addition of NQC corrections (dash-dotted line, blue online) nor of QNM ringdown (dashed line, red online). The vertical (light) dashed line indicates the location of the maximum of $M\Omega$.

Figure 4

Time evolution of the phase difference $\Delta {\varphi }_{\ell m}^{\mathrm{EOBRWZ}}={\varphi }_{\ell m}^{\mathrm{EOB}}-{\varphi }_{\ell m}^{\mathrm{RWZ}}$ between the full EOB and RWZ multipolar waveforms. The dash-dotted vertical line locates the light ring.

Figure 5

Time evolution of the relative amplitude difference $(\Delta A/A{)}_{\ell m}=(A_{\ell m}^{\mathrm{EOB}}-A_{\ell m}^{\mathrm{RWZ}})/A_{\ell m}^{\mathrm{RWZ}}$ between the full EOB and RWZ multipolar waveforms. The dash-dotted vertical line locates the light ring.

Figure 6

Comparison between the RWZ and EOB complete gravitational waveforms taken in the direction $(\theta ,\Phi )=(0,\pi /4)$. The two panels show the time evolution of the two polarizations $(h_{+},h_{\times })$ computed including up to $\ell =4$ multipoles for $\nu ={10}^{-3}$ (the $m=0$ multipoles are neglected).

Figure 7

Time evolution of the phase difference $\Delta {\varphi }^{\mathrm{EOBRWZ}}={\varphi }^{\mathrm{EOB}}-{\varphi }^{\mathrm{RWZ}}$ between the EOB and RWZ waveforms of Fig. 6. The top panel displays the behavior of this function from the very beginning (corresponding to initial separation $r=7M$); the bottom panel focuses on the late-time behavior.

Figure 8

Improvement of the EOB $\ell =2$ ringdown waveform when matching the QNMs at $t_{\mathrm{match}}=t_{\mathrm{LR}}+3M$. As in Fig. 3, the (light) dashed lines refer to the bare insplunge waveform, without the addition of NQC corrections (dash-dotted line, blue online) nor of QNM ringdown (dashed line, red online). The vertical dashed line locates the maximum of $M\Omega$.

Figure 9

Improvement of the EOB ringdown waveform when matching the QNMs at $t_{\mathrm{match}}=t_{\mathrm{LR}}+3M$. Top panels: $h_{+}$ and $h_{\times }$ GW polarizations. Bottom panel: Phase difference. Contrast it with Figs. 6, 7. The dashed-dotted vertical lines indicates the light-ring crossing.

Figure 10

Maxima of the modulus of the RWZ master function for different multipoles. The dashed lines are obtained by fitting the data with Eq. (b1).